Related papers: Elliptic Euler-Poisson-Darboux equation, critical …

We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are…

Given a smoothly bounded non-contractible domain $\Omega\subset \mathbb{R}^2$, we prove the existence of positive critical points of the Trudinger-Moser embedding for arbitrary Dirichlet energies. This is done via degree theory, sharp…

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to…

Second order divergence form operators are studied on an open set with various boundary conditions. It is shown that the p-ellipticity condition of Carbonaro-Dragicevic and Dindos-Pipher implies extrapolation to a holomorphic semigroup on…

Using the free fermions technique and bosonization rules we introduce the multicomponent DKP hierarchy as a generating bilinear integral equation for the tau-function. A number of bilinear equations of the Hirota-Miwa type are obtained as…

We study the derivation of ion dynamics, namely, the ionic Euler--Poisson system, from kinetic descriptions. The kinetic framework consists of the ionic Vlasov--Poisson equation coupled with either a nonlinear Fokker--Planck operator or a…

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1, 2}\cap L^{\infty}(\Omega;\mathbb{R}^N)$) of the vector-valued, Euler-Lagrange system: \text{div } \big( A(x, u)Du\big)=g(x,…

We provide results on the existence, non-existence, multiplicity and localization of positive radial solutions for semi linear elliptic systems with Dirichlet or Robin boundary conditions on an annulus. Our approach is topological and…

We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the…

A class of nonlinear problems on the plane, described by nonlinear inhomogeneous $\bar{\partial}$-equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources) is described…

We develop a new approach to the classification of integrable equations of the form $$ u_{xy}=f(u, u_x, u_y, \triangle_z u \triangle_{\bar z}u, \triangle_{z\bar z}u), $$ where $\triangle_{ z}$ and $\triangle_{\bar z}$ are the…

We investigate the properties of certain elliptic systems leading, a~priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution…

Conservation laws, heirarchies, scattering theory and B\"acklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply…

The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a $q$-deformation of the Ehrhart series, based on the notions of harmonic spaces and Macaulay's inverse…

Interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systInterpretation of dispersionless integrable hierarchies as equations…

We propose elliptical graphical models based on conditional uncorrelatedness as a general- ization of Gaussian graphical models by letting the population distribution be elliptical instead of normal, allowing the fitting of data with…

It is shown that the dispersionless scalar integrable hierarchies and, in general, the universal Whitham hierarchy are nothing but classes of integrable deformations of quasiconformal mappings on the plane. Examples of deformations of…

In this paper we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of "times". Our…

We study Green's matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of the Green's matrices.

We deal with boundary value problems for second-order nonlinear elliptic equations in divergence form, which emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of…